Oplatek

It's interesting that the first meaning morphs into another, but both make sense.  Many things catholic are "both" and "and".

The old custom is that the pope breaks off a piece for his bishops, who in turn bring it to their home diocese, where they break off a piece for their priests.  

Also, watching the Mass celebrant break off a piece of the Body, and drop it into the Blood, unite the two into One.  I think that's a beautiful image.

It is also reminiscent of my own Lithuanian heritage.  Where the father of the family takes unconsecrated communion and breaks off a piece and passes it on, to the rest of the family.